Asymptotic expansions and scaling of turbulent boundary layers in adverse pressure gradients

This paper presents an outer velocity scale and an inner scaling law of Reynolds shear stresses that are valid for the entire adverse pressure gradient (APG) region before separation, based on asymptotic expansions in turbulent boundary layers (TBLs). By reexamining the velocity defect and Reynolds shear stress in the outer region of TBLs, it is demonstrated that the pressure gradient effect can be included in asymptotic expansions. The mathematical requirements to be satisfied by the outer velocity scale are given. On this basis, we define a new outer velocity scale U-o, which is used in the mean momentum balance analysis over the entire APG region. The results show that there is a slowly evolving region of the Reynolds shear stress gradient in APG TBLs. We obtain a logarithmic decay law for the Reynolds stress gradient by asymptotic matching in this region, thus providing functional expressions for the total shear stress and total shear stress gradient in the inner region of TBLs. These expressions are valid for the entire APG region. Furthermore, we use the new velocity scale to determine three self-similar parameters in the momentum equation to discuss the conditions for approximate equilibrium. The changes in parameters point to changes in two physical properties of APG TBLs, the inertial force and turbulence production.